Finite Rank Perturbations and Model Theory

许多物理系统都是用微分算子来建模的。描述系统长期行为的一种方法是通过频谱理论，它包括寻找系统自然表现出来的频率。想象一根固定长度的振动弦或梁。显然，它的频率(想想“声音”)取决于它的末端(又名边界)是如何被限制或以其他方式限制的。通常，当施加在边界上的条件改变时，微分算符的谱以及物理系统的性质会发生巨大的变化。在许多情况下，我们知道一组边界条件的完整谱。从那里，我们可以通过所谓的有限秩扰动理论获得关于系统在其他条件下的知识。该项目的主要目标是通过加强与相应函数模型的关系来系统地研究有限秩酉扰动谱理论。排名第一的设置是相当好理解的，而有限排名问题的一般处理则提出了几个离题。最初，非循环酉无扰算子提出了一个问题。这个问题现在已经解决了。一般的问题涉及矩阵取值的赫格尔茨变换和柯西型变换。作为这个项目的一部分，对这些变换的研究将提供对有限秩扰动理论的洞察。计划了所谓的外柯西变换的正则化(如在第一阶设置中所做的那样)。我们将进一步研究迹类中具有亏损算子的无限秩扰动这一困难情况。类似的自伴设置，以及对微分算子的具体应用，也将被研究。

Many physical systems are modeled by differential operators. One way of describing a system's long-term behavior is through spectral theory, which includes the finding of frequencies naturally exhibited by the system. Imagine a vibrating string or beam of fixed length. Clearly, its frequency (think "sound") depends on how its ends (aka boundaries) are clamped down or otherwise restricted. In general, the spectrum of a differential operator and with it the properties of a physical system can change drastically when the conditions imposed on the boundary are changed. In many cases, we know the complete spectrum for one set of boundary conditions. From there, we can gain knowledge about the system under other conditions via the theory of so-called finite rank perturbations. The primary goal of the project is to systematically study spectral theory of finite rank unitary perturbations by tightening the relationship to corresponding functional models. The rank one setting is reasonably well-understood, while a general treatment of the finite rank problem presented several digressions. Initially, non-cyclic unitary unperturbed operators posed an issue. This problem is now resolved. The general problem involves matrix-valued Herglotz and Cauchy-type transforms. The study of these transforms as part of this project will provide insight into finite rank perturbation theory. A regularization of the so-called exterior Cauchy transform (as was done in the rank one setting) is planned. Further investigations will be made into the difficult case of infinite rank perturbations with defect operators in the trace class. The analogous self-adjoint setting, as well as concrete applications to differential operators, will also be investigated.